5.7 Implication 5: No Correlation on Columns (Test for independence)
Calculate Pearson correlation for every pair of columns with at least 3 LDFs
This is a test on the LDFs (Mack assumption 1 (prop. 4.1), LDFs are not correlated)
Test with only 2 LDFs will either be 1 or -1
Not just adjacent LDF pairs like in Mack-1994
\(\therefore\) For a \(m \times m\) triangle we have \({7 \choose 2}\) pairs (7 because only 9 columns of LDFs and we take out the 2 columns with less than 3 LDFs)
Correlation:
\[\begin{equation} r = \dfrac{\sum {\tilde{x}} {\tilde{y}} }{\sqrt{\sum {\tilde{x}}^2 \sum {\tilde{y}}^2}} \tag{5.4} \end{equation}\]Where \(\tilde{x} = x - \bar{x}\) and \(\tilde{y} = y - \bar{y}\)
\(x\)’s and \(y\)’s are incremental LDFs - 1
- But actually not necessary since -1 doesn’t affect the correlation
Use calculator data table for \(r\) once you have \(\tilde{x}\) and \(\tilde{y}\)
Test statistics for significance is \(T \sim t_{n-2}\)
\[\begin{equation} T = r \sqrt{ \dfrac{n-2}{1-r^2} } \tag{5.5} \end{equation}\]Look up the t-value from table for 90%
\(n\) is the number of LDF pairs
If \(|T| <\) table value \(\Rightarrow\) Not correlated
Perform test for all columns
- We deem the triangle have significant correlations if the number of correlated pairs are more than
- x = # of pair tested for a \(m \times m\) triangle